Integer sequence uniformly distributed mod $n $ for every $n $ I have the following conjecture:
Let $(x_n) $ be a strictly increasing  sequence of natural numbers satisfying the following property:
For every integer $m\geq 2$, for all $0\leq a,b\leq m -1$ and for all $k\in \mathbb N $: $$|N_m(a, k )-N_m(b, k )|\leq C (m)$$
where $N_m  (a,k ) := \#\{i\leq k \mid x_i \equiv a \pmod m\}$  and $C (m) $ is a constant depending only on $m $.
Then $x_n $ contains all the natural numbers with finitely many exceptions.
I have no idea how to prove it. 
 A: This isn't true. We can start out by omitting (let's say) $2,3$ from our sequence. As far as distribution $\bmod 2$ is concerned, once we're past this block, it has no further effect.
Next, we omit a block of length $6=2\cdot 3$, and we also position this so that distribution $\bmod 2$ isn't negatively affected. For example, $6,\ldots, 11$ would work (but we don't want to let it start at $7$, say, because then the connecting piece $4,5,6$ wouldn't have uniform distribution $\bmod 2$).
Now the same remarks apply to distribution $\bmod 2$ and $\bmod 3$. In general, at step $n$, we remove a block of length $\textrm{lcm}(1,2,3, \ldots , n)$, which we also position such that distribution $\bmod 2,\ldots , n-1$ is uniform to the left of this block.
For a given $m$, the distribution $\bmod m$ will be uniform once we're past block $m$, and the initial piece gives a bounded perturbation that will be absorbed by $C(m)$.
A: The sequence A266334 on OEIS is a counterexample to the conjecture.
It is all the numbers $n$ which satisfy $k! \leq n < 2 (k!) $ for some $ k\in \mathbb N$, in ascending order.
[Inspired by the other answer.]
